Finite Group Databasebeta

← #72254260

Subgroups — #72254260

Order 4704.

818 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72254259C212yesmaximal, complemented
2#72254258C212yesmaximal, complemented
3#72254257C212yesmaximal, complemented
4C7×#66669C212yesmaximal, complemented
5#72246605C211yesmaximal, characteristic, complemented
6Dic3×D72C212yesmaximal, complemented
7C3×#11716106C211yesmaximal, characteristic, complemented
8C4×D3×C7⋊D7C211yesmaximal, characteristic, complemented
9#72222125C212yesmaximal, complemented
10#13096688—31nomaximal, Hall
11C2×D3×C7⋊D7C2211yescharacteristic
12C2×C3×D72C2212yes—
13C4×C3⋊(C7⋊D7)C2211yescharacteristic, complemented
14C7×Dic42C2212yescomplemented
15C4×D3×C72C2211yescharacteristic, complemented
16C3×C4×C7⋊D7C2211yescharacteristic
17C3×C7×D28C2212yescomplemented
18#11924255C2212yescomplemented
19#11924252C2212yes—
20#11924249C2214yescomplemented
21D7×Dic21C2214yescomplemented
22#11924239C2211yescharacteristic, complemented
23D3×C7⋊Dic7C2211yescharacteristic, complemented
24Dic3×C7⋊D7C2211yescharacteristic
25C7×#4245C2214yescomplemented
26C7×Dic3×D7C2214yescomplemented
27C3×#87640C2212yescomplemented
28C2×C4×C7⋊D7—31no—
29C7×#1157—32no—
30#11716115—32no—
31#11716109—32no—
32#11716106D311yescharacteristic, complemented
33C4×D72—32no—
34#11716102—32no—
35#11716101—31no—
36#11716100—32no—
37D7×#237—72nomaximal
38C2×C3⋊(C7⋊D7)C2311yescharacteristic
39C2×D3×C72C2311yescharacteristic
40C2×C3×C7⋊D7C2311yescharacteristic
41C2×C3×C7×D7C2314yes—
42D3×C7⋊D7—22no—
43C3×D72—24no—
44C3×C4×C72C2311yescharacteristic
45C3⋊(C7⋊Dic7)C2311yescharacteristic
46C7×Dic21C2314yes—
47C72×Dic3C2311yescharacteristic
48C3×C7⋊Dic7C2311yescharacteristic
49C22×C7⋊D7—31no—
50C2×D72C2×D312yes—
51C2×C4×C72—31no—
52C2×C7⋊Dic7—31no—
53C4×C7⋊D7—31no—
54C4×C7⋊D7—31no—
55C4×C7⋊D7—31no—
56C4×C7⋊D7C2×D311yescharacteristic
57C7×C22⋊D7—34no—
58C7×D28C2×D312yescomplemented
59C4×C7×D7—34no—
60C7×Dic14—32no—
61#87642—32no—
62#87641—34no—
63#87640C2×D312yescomplemented
64#87639—32no—
65D7×Dic7—34no—
66#66730—72no—
67C7×#237—72no—
68C3×D4×D7—72no—
69D7×#67—74no—
70#66684—74no—
71#66683—74no—
72C2×Dic3×D7—74no—
73C4×D3×D7—72no—
74C4×D3×D7—143no—
75#66670—72no—
76#66669D712yescomplemented
77D7×Dic6—72no—
78C2×C3×C72C2411yescharacteristic
79C3⋊(C7⋊D7)—21no—
80D3×C72—21no—
81C3×C7⋊D7#4112yescomplemented
82C3×C7×D7—24no—
83D7×#41—212no—
84C22×C72—31no—
85C2×C7⋊D7—31no—
86C2×C7⋊D7—62no—
87C2×C7⋊D7C22×D311yescharacteristic
88C2×C7×D7C22×D314yes—
89D72—24no—
90C4×C72—31no—
91C4×C72C22×D311yescharacteristic
92C7⋊Dic7—31no—
93C7⋊Dic7C22×D311yescharacteristic
94C7×Dic7—34no—
95C22×C3×D7—74no—
96C2×D3×D7—72no—
97C2×D3×D7—143no—
98C3×C7×D4—72no—
99#4268—74no—
100C2×Dic21—74no—
101C4×D21—72no—
102C4×D21—143no—
103Dic42C2×D712yescomplemented
104C7×#67—74no—
105C2×C7×Dic3—74no—
106C4×D3×C7—23no—
107C4×D3×C7C2×D712yescomplemented
108C7×Dic6—72no—
109C3×C22⋊D7—74no—
110C3×D28C2×D712yescomplemented
111C3×C4×D7—72no—
112C3×C4×D7—143no—
113#4248—74no—
114#4246—74no—
115#4245C2×D714yescomplemented
116D21⋊C4—72no—
117D21⋊C4—143no—
118D3×Dic7—72no—
119D3×Dic7—143no—
120Dic3×D7—74no—
121Dic3×D7—74no—
122Dic3×D7—72no—
123Dic3×D7—143no—
124Dic3×D7C2×D714yescomplemented
125C3×C72C2×#4111yescharacteristic, complemented, Hall
126C7×#41—212no—
127#1161—212no—
128Q8×D7—212no—
129#1159—212no—
130#1159—214no—
131D4×D7—72no—
132D4×D7—214no—
133#1157—32no—
134#1157—214no—
135C2×C4×D7—214no—
136C2×C4×D7—212no—
137C2×C4×D7—423no—
138C2×C72—61no—
139C2×C72C23×D311yescharacteristic
140C7⋊D7—61no—
141C7⋊D7#23712yescomplemented
142C7×D7—24no—
143C2×#237—491noHall
144C22×C3×C7—74no—
145C2×D21—72no—
146C2×D21—143no—
147C2×D3×C7—23no—
148C2×D3×C7C22×D712yes—
149C2×C3×D7—72no—
150C2×C3×D7—143no—
151C2×C3×D7—148no—
152C2×C3×D7C22×D714yes—
153D3×D7—144no—
154D3×D7—1412no—
155C3×C4×C7—23no—
156C3×C4×C7C22×D712yes—
157Dic21—72no—
158Dic21—143no—
159Dic21C22×D714yes—
160C3×Dic7—72no—
161C3×Dic7—143no—
162C7×Dic3—23no—
163C7×Dic3—74no—
164C7×Dic3C22×D712yes—
165C22×D7—74no—
166C22×D7—212no—
167C22×D7—423no—
168C7×Q8—212no—
169C7×D4—72no—
170C7×D4—214no—
171C2×C4×C7—32no—
172C2×C4×C7—63no—
173C2×C4×C7—214no—
174C22⋊D7—34no—
175C22⋊D7—74no—
176C22⋊D7—214no—
177C2×Dic7—214no—
178C2×Dic7—212no—
179C2×Dic7—423no—
180D28—214no—
181D28D3×D712yescomplemented
182C4×D7—34no—
183C4×D7—72no—
184C4×D7—143no—
185C4×D7—214no—
186C4×D7—212no—
187C4×D7—212no—
188C4×D7—214no—
189C4×D7—212no—
190C4×D7—423no—
191C4×D7—423no—
192C4×D7—423no—
193Dic14—32no—
194Dic14—214no—
195C72C2×#23711yescharacteristic, complemented, Sylow
196C2×C3×D4—491no—
197C2×#67—492no—
198C22×Dic3—492no—
199#237—72no—
200#237—492no—
201#237—492no—
202#237—492no—
203C2×C4×D3—491no—
204C2×Dic6—491no—
205C2×C3×C7—23no—
206C2×C3×C7—144no—
207C2×C3×C7C23×D712yes—
208D21—146no—
209D21—142no—
210C3×D7—24no—
211C3×D7—74no—
212C3×D7—146no—
213D3×C7—26no—
214D3×C7—22no—
215C2×#41—1471noSylow
216C22×C7—32no—
217C22×C7—63no—
218C22×C7—74no—
219C2×D7—72no—
220C2×D7—143no—
221C2×D7—148no—
222C2×D7—212no—
223C2×D7—424no—
224C2×D7—4212no—
225C2×D7—423no—
226C2×D7C2×D3×D714yes—
227C4×C7—23no—
228C4×C7—32no—
229C4×C7—63no—
230C4×C7—214no—
231C4×C7C2×D3×D712yes—
232Dic7—34no—
233Dic7—72no—
234Dic7—143no—
235Dic7—212no—
236Dic7—423no—
237C23×C3—492no—
238C22×D3—491no—
239C3×D4—72no—
240C3×D4—492no—
241C2×C3×C4—491no—
242#67—74no—
243#67—494no—
244C2×Dic3—74no—
245C2×Dic3—492no—
246C2×Dic3—494no—
247C2×Dic3—491no—
248C4×D3—491no—
249C4×D3—491no—
250C4×D3—491no—
251C4×D3D7211yescharacteristic, complemented
252Dic6—72no—
253Dic6—492no—
254C3×C7—23no—
255C3×C7D7×#4112yescomplemented
256#41—212no—
257#41—1472no—
258#41—1472no—
259#41—1472no—
260C2×Q8—1471no—
261C2×D4—491no—
262C2×D4—1472no—
263C22×C4—1471no—
264C22×C4—1472no—
265C2×C7—23no—
266C2×C7—66no—
267C2×C7—62no—
268C2×C7—144no—
269C2×C7C22×D3×D712yes—
270D7—24no—
271D7—74no—
272D7—146no—
273D7—426no—
274D7—422no—
275C22×C3—74no—
276C22×C3—491no—
277C22×C3—984no—
278C2×D3—491no—
279C2×D3—982no—
280C2×D3C2×D7211yescharacteristic
281C3×C4—491no—
282C3×C4C2×D7211yescharacteristic
283Dic3—74no—
284Dic3—491no—
285Dic3C2×D7211yescharacteristic
286C23—492no—
287C23—1471no—
288Q8—212no—
289Q8—1472no—
290D4—72no—
291D4—214no—
292D4—492no—
293D4—1474no—
294C2×C4—31no—
295C2×C4—214no—
296C2×C4—491no—
297C2×C4—1471no—
298C2×C4—1471no—
299C2×C4—1472no—
300C2×C4—1474no—
301C2×C4—1471no—
302C2×C4—1471no—
303C7—23no—
304C7D7×#23712yescomplemented
305C2×C3—144no—
306C2×C3—492no—
307C2×C3C22×D7211yescharacteristic
308D3—21no—
309D3—981no—
310C22—31no—
311C22—74no—
312C22—2942no—
313C22—491no—
314C22—984no—
315C22—1471no—
316C4—31no—
317C4—214no—
318C4—491no—
319C4—1471no—
320C4D3×D7211yescharacteristic
321C3#1309668811yescharacteristic, complemented, Sylow
322C2—61no—
323C2—144no—
324C2—2941no—
325C2—492no—
326C2C2×D3×D7211yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×D72
commutator, socle, cosocleC2×C3×C72C24
FittingC3×C4×C72C23
hypercenterC4D3×D72
hypocenterC3×C72C2×#41
LCSG ▷ C2×C3×C72 ▷ C3×C72
UCSC1 ◁ C2 ◁ C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #72254259C2semidirect
2. #72254258C2semidirect
3. #72254257C2semidirect
4. C7×#66669C2semidirect
5. #72246605C2semidirect
6. Dic3×D72C2semidirect
7. C3×#11716106C2semidirect
8. C4×D3×C7⋊D7C2semidirect
9. #72222125C2semidirect
13. C4×C3⋊(C7⋊D7)C22semidirect
14. C7×Dic42C22semidirect
15. C4×D3×C72C22semidirect
17. C3×C7×D28C22semidirect
18. #11924255C22semidirect
20. #11924249C22semidirect
21. D7×Dic21C22semidirect
22. #11924239C22semidirect
23. D3×C7⋊Dic7C22semidirect
25. C7×#4245C22semidirect
26. C7×Dic3×D7C22semidirect
27. C3×#87640C22semidirect
32. #11716106D3semidirect
58. C7×D28C2×D3semidirect
63. #87640C2×D3semidirect
76. #66669D7semidirect
81. C3×C7⋊D7#41semidirect
103. Dic42C2×D7semidirect
107. C4×D3×C7C2×D7semidirect
110. C3×D28C2×D7semidirect
115. #4245C2×D7semidirect
124. Dic3×D7C2×D7semidirect
125. C3×C72C2×#41semidirect
141. C7⋊D7#237semidirect
181. D28D3×D7semidirect
195. C72C2×#237semidirect
251. C4×D3D72semidirect
255. C3×C7D7×#41semidirect
304. C7D7×#237semidirect
321. C3#13096688semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#722542592
#722542582
#722542572
C7×#666692
#722466052
Dic3×D722
C3×#117161062
C4×D3×C7⋊D72
#722221252
#130966883
D7×#2377